Theorem plendxnmulrndx

Description: The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr . (Contributed by AV, 1-Nov-2024)

	Ref	Expression
Assertion	plendxnmulrndx	⊢ ( le ‘ ndx ) ≠ ( .r ‘ ndx )

Proof

Step	Hyp	Ref	Expression
1		3re	⊢ 3 ∈ ℝ
2		3lt10	⊢ 3 < ; 1 0
3	1 2	gtneii	⊢ ; 1 0 ≠ 3
4		plendx	⊢ ( le ‘ ndx ) = ; 1 0
5		mulrndx	⊢ ( .r ‘ ndx ) = 3
6	4 5	neeq12i	⊢ ( ( le ‘ ndx ) ≠ ( .r ‘ ndx ) ↔ ; 1 0 ≠ 3 )
7	3 6	mpbir	⊢ ( le ‘ ndx ) ≠ ( .r ‘ ndx )